A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Book recommendation needed: asymptotic behavior of non-stationary Markov chain

Is there any stochastic process textbook which covers some standard results for non-stationary Markov chain? For my purpose, countable state space is enough. Thanks!
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6 views

How Solve these expectations

Let $T$ be the exit time of from the interval $[-b,a]$ of a standard Brownian Motion $X_t$, then how would we go about calculating the following two expectations: $E[T^2]$ (and) $E[\int_0^T X_tds]$? ...
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Measurability of the event that Brownian motion hits a given set

Let $W$ be a Brownian motion in $\mathbb{R}^{2}$ on a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ . Let us assume $\mathcal{F}$ is the sigma-algebra on the path space ...
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5 views

Point processes that are not Cox?

Can some provide examples of point processes that are not Cox? A Cox process is a doubly stochastic poisson process with random intensity.
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15 views

Applying stopping times to obtain a conditional probability

Let $M$ be a positive, continuous martingale that converges a.s. to zero as $t$ tends to infinity. I now want to prove that for every $x>0$ $$ P\left( \sup_{t \geq 0 } M_t > x | \mathcal{F}_0 ...
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17 views

Supremal distribution of positive continuous martingale, which converges to zero a.s.

So the question is as follows: Let $M$ be a positive continous martingale, converging a.s. to zero as $t \rightarrow \infty$. Prove that for every $x>0$: $\mathbb{P}\{\sup_{\{t \geq 0 \}} M_t > ...
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9 views

Relationship between distributions of correlations $\rho(X^1,Y^1)$ and $\rho(X^2,Y^2)$ if $X^2=WX^1$, $Y^2=WY^1$ and $W$ is a known stochastic matrix?

I have been stacked for a while with the following problem: Consider two samples of iid observations $X^1=\{X_1^1,\dots,X_n^1\}$ and $Y_1=\{Y_1^1,\dots,Y_n^1\}$ where $X_i^1 \sim \mathcal{N}(0,1)$ and ...
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1answer
23 views

Reference request for stochastic process

I studied the book, "Probability with the book, Probability, Random Variables and Random Signal Principles" by Peyton Peebles. And I am a little bit familiar with statistical analysis like signal ...
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1answer
29 views

If two stochastic processes are modifications of each other and almost surely continuous from the right, then they are undistinguishable

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $I\subseteq\mathbb{R}$ $E$ be a metric space and $\mathcal{E}:=\mathcal{B}(E)$ be the Borel-$\sigma$-algebra on $E$ $X:=(X_t)_{t\in ...
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14 views

Defining a stochastic process indexed by angle

I have a random closed curve of the form $(\theta,r_\theta)$, where $\theta\in [0,2\pi]$, is the counter clockwise angle from the x-axis and $r_\theta$ is the radial distance from the origin ...
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1answer
33 views

Adaptive gambler's ruin problem

Suppose in the gambler's ruin problem that the probability of winning a bet depends on the gambler's present fortune. Specifically, suppose that $p_{i}$ is the probability that the gambler wins a bet ...
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1answer
25 views

Men and Women enter a supermarket according to independent poisson process (stochastic process) [on hold]

Men and Women enter a supermarket according to independent poisson processes having respective rates of two and four per minute. a) Starting at an arbitrary time, what is the probability that at ...
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12 views

Variance of interarrival time of events [on hold]

As shown in the figure, in this problem, there are three types of events where events of each type occur independently. The inter-arrival time distribution between events of the same type is an ...
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1answer
52 views

Generating the Borel $\sigma$-algebra on $C([0,1])$

We put $S=C([0,1])$ (the collection of continuous real functions on $[0,1]$), equipped with the metric $d(f,g)=\sup_{x\in[0,1]}|f(x)-g(x)|$, and let $\mathcal{B}(S)$ be the Borel $\sigma$-algebra on ...
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2answers
47 views

How to prove $E[e^{e^y}]=\infty$? y is a normal random variable

The question is, given $Y\sim N(\mu,\sigma^2)$, how to prove$E[e^{e^Y}]=\infty$? I tried to look Y as some kind of Ito's process and apply Ito's formula to it but it doesn't make sense. Next I tried ...
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1answer
31 views

Distribution of $\| W_t \|^2_{L^2([0,T])}$

Motivation: consider the SDE $$dX_t = b(X_t) dt + \sqrt{\varepsilon} dW_t. \tag1$$ Consider the action, defined by $$S(\phi)=\int_0^T |\phi'(t)-b(\phi(t))|^2 dt$$ if $\phi \in H^1([0,T])$ and ...
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1answer
48 views

Show that a Markov Chain is ergodic

Let $Y_n$ be iid random variables with values 1,2,3..n so that $P[Y_i=j]=p_j>0$, where $i\leq1$ and $1\leq j\leq n$. I think I managed to show that $Y_n$ is a Markov chain using the definition, ...
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1answer
34 views

Stronger version of Markov Chain

I have just started looking into the concept of Markov chains and I was wondering if anyone could help me with this problem. Let $X_1, X_2, ...$ be a Markov chain with the state space $S$. I need ...
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1answer
29 views

Process adapted to Filtration [on hold]

Here is the definition I have been given : A process $(X_t)$ is adapted to a filtration $(\mathcal F_t)$ if $X_t$ is $F_t$ measurable, for all t > 0 , i.e : $X_t^-1 (\mathcal B)$ belong to ...
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Convergence in distribution of BM started in (x,y) to BM started in (0,0)

Let $B$ be a Brownian motion in $\mathbb{R}^{2}$ . Let $\mathbb{P}_{(x,y)}$ denote a probability measure under which $B$ is started at $(x,y)$ . Is it true in general that, for measurable set ...
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21 views

Simple Symmetric Random Walk [on hold]

Use Hint: Show first that for any random variable N with range {0,1,...},
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1answer
25 views

Example of a white noise series that is not a martingale difference series with respect to its natural filtration

For a homework exercise, I am asked to find an example of a white noise series that is not a martingale difference series with respect to its natural filtration. Does anyone know an example? I read ...
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17 views

How to get the PDF for $p(a)$ $p(b)$ by using convolution?

Suppose there are three random variable $a$, $b$, $c$, and the PDF for each are $p(a)\ p(b)$ and $p(c) $ Also, $c$ = $a$ + $b$, $a$ and $b$ are two independent variable. and$$p(c)= \begin{cases} ...
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1answer
28 views

Applying the martingale representation theorem

I'm having trouble applying the martingale representation theorem to examples of Brownian martingales $M$ and contruct a process $X$ such that if we have a Brownian motion $W$ then $M= X \cdot W$. ...
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1answer
26 views

Stochastic kernel as linear operator

Let $K$ be a stochastic kernel for a set $S$ equipped with a countably generated $\sigma$-Algebra $B(S)$, i.e. $K:S\times B(S)\rightarrow [0,1]$ such that $K(\cdot,A)$ is a measurable function for ...
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1answer
19 views

Finding the expectation and variance of a stochastic process

Let $X_0, \ldots$ be i.i.d. $\mathbb{P}\{X_i = -1\} = \mathbb{P}\{X_i = 1\} = 1 / 2$. Given $a, b \in \mathbb{R}, |b| < 1$, consider the stochastic process $W_k$ defined as $$ W_0 = a X_0\\ W_k = b ...
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25 views

Two-state Markov Chains

If I have a two-state Markov chian $V(t)$ with transition probabilities: $P_{00}(t)=(1-\pi) + \pi e^{-\tau t}$ $P_{01}(t)= \pi - \pi e^{-\tau t}$ $P_{10}(t)=(1-\pi) - (1-\pi)e^{-\tau t}$ ...
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29 views

Does the power spectral density vanish when the frequency is zero for a zero-mean process?

A wide-sense stationary random time series $\zeta(t)$ is characterized by its mean value and its autocovariance function, which in the Wiener–Khinchin theorem is equivalent to the Fourier transform of ...
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55 views

Find the expected value ${\mathbb E}(x_k)$

$X_k = X_{k-1} + m_k(x_0=0)$, and pdf(probability density function) of $m_k$ is defined as $$p(m_k)= \begin{cases} m_k+1&-1\le m_k<0\\ -m_k+1&0\le m_k\le1\\ 0&\text{otherwise} ...
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20 views

Finding solution to this stochastic differential equation

Let $W, Z$ be two correlated Brownian motions with $dW\,dZ=\rho\, dt$. We also have the following three processes: \begin{align} dD_t &= rD_t \,dt & & (D_T=1, r>0)\\ dS_t &= rS ...
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Is there an analytic solution for this Fokker-Planck equation?

The Fokker-Planck equation for a probability distribution $P(\theta,t)$: \begin{align} \frac{\partial P(\theta,t)}{\partial ...
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1answer
21 views

Using Central Limit Theorem to show that random walk exits a interval a.s. in finite time.

Let $X_0 = x \in \mathbb{Z}$ and $X_1, X_2, \dots$ are i.i.d. random variables with values in $\{-1,0,1\}$ all with positive probability and $E(X_1) = 0$. Let $\sigma^2 = E(X_1^2)$. Let $S_n = ...
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57 views

Right-continuous supermartingale is almost sure cadlag

Suppose that we have a supermartingale $M$ defined on an underlying filtered probability space $(\Omega,\mathcal{F},P)$. We assume that the index set is defined by $T=\mathbb{R}^+$ and for notation we ...
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28 views

how to determine transient and recurrent state from transition matrix

I wonder how can I determine the transient and recurrent state from transition matrix ? I mean if I have 10 states It would be very hard to draw diagram for them so how to analyse the matrix? For ...
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1answer
29 views

Calculating inter-arrival times and arrival times of a Poisson process

For a practice exam in stochastic processes I have to answer the following questions. Let $\{N(t): t\geq 0\}$ be a poisson process with rate $\lambda$. Let $T_n$ denote the n-th inter-arrival time ...
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1answer
32 views

Weak convergence of an integral of an exponential of a Wiener process

Suppose $(W_s)_{s \geq 0}$ is a Wiener process. Define $$ V_t := \frac{1}{\sqrt{t}}\int_{0}^{t}\exp(W_s)ds $$ Show that $$ V_t \xrightarrow{t \rightarrow \infty} \sup_{s \in[0, 1]}W_s $$ in ...
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1answer
15 views

Stationary Probabilities: Periodic Case: motivation 2nd attempt.

For DTMC with $S=\{1,2\}$ and transition probabilities $$P = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$ How do we see that $(P_{00})^{(n)} = 1$ if $n$ is even or $0$ if $n$ is odd ?? ...
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1answer
27 views

Stationary probabilities: periodic case: motivation

For DTMC with $S=\{1,2\}$ and transition probabilities $$P = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$ How do we see that $(P_{00})^{(n)} = 1$ if $n$ is even or $0$ if $n$ is odd ?? ...
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35 views

Lindeberg-Feller CLT follows from Martingale CLT?

I've been studying about central limit theorems, in particular the Lindeberg-Feller CLT and other extensions. In most textbooks and sources online the martingale central limit theorem comes as an ...
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1answer
28 views

Kullback-Leibler divergence when the $Q$ distribution has zero values

For discrete probability distributions $P,Q$, the Kullback-Leibler divergence of $Q$ from $P$ is defined to be $$D_{\mathrm{KL}} ( P \mathop{\|} Q ) = \sum_i P(i) \ln \left( \frac{P(i)}{Q(i)} ...
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1answer
19 views

Period of a Markov Chain: Why is this one aperiodic?

Here is the problem from a stochastic processes book: Consider a Markov Chain on {0,1,2} having transition matrix 0 1 2 0| 0 0 1| 1| 1 0 0| 2|.5 .5 0| ...
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24 views

Optimization of stochastic differential equations

Is there a way to optimize or maximize a set of differential equations. such that each equation is represented by a time series S_((t+1),μ) = μ*(S_(t+1)-S_t) + S_t and μ = 2/(i+1), i=1,...,n. Then I ...
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1answer
34 views

Joint Density Function of uniform and gamma density [closed]

Let $U$ be uniformly distributed over the interval $[0, L]$ where $L$ follows the gamma density $f_L(x) = xe^{-x}$ for $x\ge 0$. What is the joint desnity function of $U$ and $V = L - U$?
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45 views

Independent Exponentially Distributed RV's [closed]

Consider a post office with two clerks. John, Paul, and Naomi enter simultaneously. John and Paul go directly to the clerks, while Naomi must wait until either John or Paul is finished before she ...
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Smoothness of marginal distribution of a diffusion in the initial condition

Let $\{p_t\}_{t \ge 0}$ be a one-dimensional diffusion process (on [0,1] ) with drift $\mu(p) = C_1(1-p)-C_2p+p(1-p)s(p),$ where $s$ is a Lipschitz function and $C_1,C_2 \in (0,1)$, and diffusion ...
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1answer
14 views

Power spectral density of the system output

$w(t)$ and $z(t)$: two stationary random processes $z(t) = Pw(t)$. $P$: a stable, LTI system. How to show: $$ S_z(jw) = P(jw)S_w(jw)P(jw)^*$$ $S_z(jw)$ is the power spectral density of ...
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1answer
25 views

Convergence of random variables in $L^1$

So $g$ is a continuous real-valued function and are given that the sequence of random variables $Y_n$ converges to $Y$ in $L^1$, $E[|g(Y_n)|]<\infty$ and $E[|g(Y)|]<\infty$. Show that $g(Y_n)$ ...
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6 views

Formula for $X_t - X_{t+h}$ where $X_t$ is $MA(q)$ process

Lets say that we have $MA(q)$ process $X_t = Z_t + \psi_1 Z_{t-1} + \dots+\psi_q Z_{t-q}$. $Z_t$ are IID (normal with mean $0$ and standard deviation $\sigma$). Now I need to find form of $X_t - ...
2
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1answer
41 views

Stochastic integral where the integrator is zero in probability

We are given a continuous semimartingale $Y$ and a continuous process $B$ of finite variation. Hence, we know that $\langle B \rangle$, the quadratic covariation of $B$, is zero in probability. I now ...
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1answer
46 views

Expectation and Convergence of Sum of Random Variables [closed]

Let $X_1, X_2, ...$ be a sequence of independent random variables with $$\mathbb{P}[X_i=1]=\mathbb{P}[X_i=-1]=\frac{1}{2}$$ Let's now consider the sum $S_n=\sum_{k=1}^{n} X_k$. I need to show three ...